MOTIVATION: Sensitivity analysis is a useful tool to identify key parameters in metabolic models. It is typically only applied to the growth rate, disregarding the sensitivity of other solution variables to parameters. Further, sensitivity analysis of elementary flux modes could provide low-dimensional insights into optimal solutions, but they are not defined when a model is subject to inhomogeneous flux constraints, such as the frequently used ATP maintenance reaction. RESULTS: We introduce optimal flux modes (OFMs), an analogue to elementary flux modes (EFMs), but specifically applied to optimal solutions of constraint-based models. Further, we prove that implicit differentiation can always be used to efficiently calculate the sensitivities of both whole-model solutions and OFM-based solutions to model parameters. This allows for fine-grained sensitivity analysis of the optimal solution, and investigation of how these parameters exert control on the optimal composition of OFMs. This novel framework is implemented in DifferentiableMetabolism.jl, a software package designed to efficiently differentiate solutions of constraint-based models. To demonstrate scalability, we differentiate solutions of 342 yeast models; additionally we show that sensitivities of specific subsystems can guide metabolic engineering. Applying our scheme to an Escherichia coli model, we find that OFM sensitivities predict the effect of knockout experiments on waste product accumulation. Sensitivity analysis of OFMs also provides key insights into metabolic changes resulting from parameter perturbations. AVAILABILITY AND IMPLEMENTATION: Software introduced here is available as open-source Julia packages DifferentiableMetabolism.jl (https://github.com/stelmo/DifferentiableMetabolism.jl) and ElementaryFluxModes.jl (https://github.com/HettieC/ElementaryFluxModes.jl), which both work on all major operating systems and computer architectures. Code to reproduce all results is available from https://github.com/HettieC/DifferentiableOFMPaper, and as an archive from https://doi.org/10.5281/zenodo.15183208.

SUMMARY: Constraint-based metabolic models offer a scalable framework to investigate biological systems using optimality principles. Construction and simulation of detailed models that utilize multiple kinds of constraint systems pose a significant coding overhead, complicating implementation of new types of analyses. We present an improved version of the constraint-based metabolic modeling package COBREXA, which utilizes a hierarchical model construction framework that decouples the implemented analysis algorithms into independent, yet re-combinable, building blocks. By removing the need to re-implement modeling components, assembly of complex metabolic models is simplified, which we demonstrate on use-cases of resource-balanced models, and enzyme-constrained flux balance models of interacting bacterial communities. Notably, these models show improved predictive capabilities in both monoculture and community settings. In perspective, the re-usable model-building components in COBREXA 2 provide a sustainable way to handle increasingly complex models in constraint-based modeling. AVAILABILITY AND IMPLEMENTATION: COBREXA 2 is available from https://github.com/COBREXA/COBREXA.jl, and from Julia package repositories. COBREXA 2 works on all major operating systems and computer architectures. Documentation is available at https://cobrexa.github.io/COBREXA.jl/.

Cyanobacteria hold great potential to revolutionize conventional industries and farming practices with their light-driven chemical production. To fully exploit their photosynthetic capacity and enhance product yield, it is crucial to investigate their intricate interplay with the environment including the light intensity and spectrum. Mathematical models provide valuable insights for optimizing strategies in this pursuit. In this study, we present an ordinary differential equation-based model for the cyanobacterium Synechocystis sp. PCC 6803 to assess its performance under various light sources, including monochromatic light. Our model can reproduce a variety of physiologically measured quantities, e.g. experimentally reported partitioning of electrons through four main pathways, O2 evolution, and the rate of carbon fixation for ambient and saturated CO2. By capturing the interactions between different components of a photosynthetic system, our model helps in understanding the underlying mechanisms driving system behavior. Our model qualitatively reproduces fluorescence emitted under various light regimes, replicating Pulse-amplitude modulation (PAM) fluorometry experiments with saturating pulses. Using our model, we test four hypothesized mechanisms of cyanobacterial state transitions for ensemble of parameter sets and found no physiological benefit of a model assuming phycobilisome detachment. Moreover, we evaluate metabolic control for biotechnological production under diverse light colors and irradiances. We suggest gene targets for overexpression under different illuminations to increase the yield. By offering a comprehensive computational model of cyanobacterial photosynthesis, our work enhances the basic understanding of light-dependent cyanobacterial behavior and sets the first wavelength-dependent framework to systematically test their producing capacity for biocatalysis.

The physiology of biological cells evolved under physical and chemical constraints, such as mass conservation across the network of biochemical reactions, nonlinear reaction kinetics, and limits on cell density. For unicellular organisms, the fitness that governs this evolution is mainly determined by the balanced cellular growth rate. We previously introduced growth balance analysis (GBA) as a general framework to model and analyze such nonlinear systems, revealing important analytical properties of optimal balanced growth states. It has been shown that at optimality, only a minimal subset of reactions can have nonzero flux. However, no general principles have been established to determine if a specific reaction is active at optimality. Here, we extend the GBA framework to study the optimality of each biochemical reaction, and we identify the mathematical conditions determining whether a reaction is active or not at optimal growth in a given environment. We reformulate the mathematical problem in terms of a minimal number of dimensionless variables and use the Karush-Kuhn-Tucker (KKT) conditions to identify fundamental principles of optimal resource allocation in GBA models of any size and complexity. Our approach helps to identify from first principles the economic values of biochemical reactions, expressed as marginal changes in cellular growth rate; these economic values can be related to the costs and benefits of proteome allocation into the reactions' catalysts. Our formulation also generalizes the concepts of Metabolic Control Analysis to models of growing cells. We show how the extended GBA framework unifies and extends previous approaches of cellular modeling and analysis, putting forward a program to analyze cellular growth through the stationarity conditions of a Lagrangian function. GBA thereby provides a general theoretical toolbox for the study of fundamental mathematical properties of balanced cellular growth.

In humans, glycogen storage diseases result from metabolic inborn errors, and can lead to severe phenotypes and lethal conditions. Besides these rare diseases, glycogen is also associated to widely spread societal burdens such as diabetes. Glycogen is a branched glucose polymer synthesised and degraded by a complex set of enzymes. Over the past 50 years, the structure of glycogen has been intensively investigated. Yet, the interplay between the detailed three-dimensional glycogen structure and the related enzyme activity is only partially characterised and still to be fully understood. In this article, we develop a stochastic coarse-grained and spatially resolved model of branched polymer biosynthesis following a Gillespie algorithm. Our study largely focusses on the role of the branching enzyme, and first investigates the properties of the model with generic parameter values, before comparing it to in vivo experimental data in mice. It arises that the ratio of glycogen synthase over branching enzyme reaction rates drastically impacts the structure of the granule. We deeply investigate the mechanism of branching and parametrise it using distinct lengths. Not only do we consider various possible sets of values for these lengths, but also distinct rules to apply them. We show how combining various values for these lengths finely tunes glycogen macromolecular structure. Comparing the model with experimental data confirms that we can accurately reproduce glycogen chain length distributions in wild type mice. Additional granule properties obtained for this fit are also in good agreement with typically reported values in the experimental literature. Nonetheless, we find that the mechanism of branching must be more flexible than usually reported. Overall, our model provides a theoretical basis to quantify the effect that single enzymatic parameters, in particular of the branching enzyme, have on the chain length distribution. Our generic model and methods can be applied to any glycogen data set, and could in particular contribute to characterise the mechanisms responsible for glycogen storage disorders.

SUMMARY: COBREXA.jl is a Julia package for scalable, high-performance constraint-based reconstruction and analysis of very large-scale biological models. Its primary purpose is to facilitate the integration of modern high performance computing environments with the processing and analysis of large-scale metabolic models of challenging complexity. We report the architecture of the package, and demonstrate how the design promotes analysis scalability on several use-cases with multi-organism community models. AVAILABILITY AND IMPLEMENTATION: https://doi.org/10.17881/ZKCR-BT30. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: Computational mathematical models of biological and biomedical systems have been successfully applied to advance our understanding of various regulatory processes, metabolic fluxes, effects of drug therapies, and disease evolution and transmission. Unfortunately, despite community efforts leading to the development of SBML and the BioModels database, many published models have not been fully exploited, largely due to a lack of proper documentation or the dependence on proprietary software. To facilitate the reuse and further development of systems biology and systems medicine models, an open-source toolbox that makes the overall process of model construction more consistent, understandable, transparent, and reproducible is desired. RESULTS AND DISCUSSION: We provide an update on the development of modelbase, a free, expandable Python package for constructing and analysing ordinary differential equation-based mathematical models of dynamic systems. It provides intuitive and unified methods to construct and solve these systems. Significantly expanded visualisation methods allow for convenient analysis of the structural and dynamic properties of models. After specifying reaction stoichiometries and rate equations modelbase can automatically assemble the associated system of differential equations. A newly provided library of common kinetic rate laws reduces the repetitiveness of the computer programming code. modelbase is also fully compatible with SBML. Previous versions provided functions for the automatic construction of networks for isotope labelling studies. Now, using user-provided label maps, modelbase v1.2.3 streamlines the expansion of classic models to their isotope-specific versions. Finally, the library of previously published models implemented in modelbase is growing continuously. Ranging from photosynthesis to tumour cell growth to viral infection evolution, all these models are now available in a transparent, reusable and unified format through modelbase. CONCLUSION: With this new Python software package, which is written in currently one of the most popular programming languages, the user can develop new models and actively profit from the work of others. modelbase enables reproducing and replicating models in a consistent, tractable and expandable manner. Moreover, the expansion of models to their isotopic label-specific versions enables simulating label propagation, thus providing quantitative information regarding network topology and metabolic fluxes.

MOTIVATION: Network-centered studies in systems biology attempt to integrate the topological properties of biological networks with experimental data in order to make predictions and posit hypotheses. For any topology-based prediction, it is necessary to first assess the significance of the analyzed property in a biologically meaningful context. Therefore, devising network null models, carefully tailored to the topological and biochemical constraints imposed on the network, remains an important computational problem. RESULTS: We first review the shortcomings of the existing generic sampling scheme-switch randomization-and explain its unsuitability for application to metabolic networks. We then devise a novel polynomial-time algorithm for randomizing metabolic networks under the (bio)chemical constraint of mass balance. The tractability of our method follows from the concept of mass equivalence classes, defined on the representation of compounds in the vector space over chemical elements. We finally demonstrate the uniformity of the proposed method on seven genome-scale metabolic networks, and empirically validate the theoretical findings. The proposed method allows a biologically meaningful estimation of significance for metabolic network properties.

A major challenge in systems biology is to understand how complex and highly connected metabolic networks are organized. The structure of these networks is investigated here by identifying sets of metabolites that have a similar biosynthetic potential. We measure the biosynthetic potential of a particular compound by determining all metabolites than can be produced from it and, following a terminology introduced previously, call this set the scope of the compound. To identify groups of compounds with similar scopes, we apply a hierarchical clustering method. We find that compounds within the same cluster often display similar chemical structures and appear in the same metabolic pathway. For each cluster we define a consensus scope by determining a set of metabolites that is most similar to all scopes within the cluster. This allows for a generalization from scopes of single compounds to scopes of a chemical family. We observe that most of the resulting consensus scopes overlap or are fully contained in others, revealing a hierarchical ordering of metabolites according to their biosynthetic potential. Our investigations show that this hierarchy is not only determined by the chemical complexity of the metabolites, but also strongly by their biological function. As a general tendency, metabolites which are necessary for essential cellular processes exhibit a larger biosynthetic potential than those involved in secondary metabolism. A central result is that chemically very similar substances with different biological functions may differ significantly in their biosynthetic potentials. Our studies provide an important step towards understanding fundamental design principles of metabolic networks determined by the structural and functional complexity of metabolites.